Lecture 12 CSE 331 Sep 28, 2012

HW 3 due today I will not take any HW after 1:15pm Q1, Q2 and Q3 in different piles I will not take any HW after 1:15pm

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Other HW related stuff HW 4 has been posted online: see blog/piazza Solutions to HW 3 at the END of the lecture Graded HW 2: pickup from TAs next week

Tree Connected undirected graph with no cycles

Rooted Tree

A rooted tree AC’s child=SG Pick any vertex as root SG’s parent=AC Let the rest of the tree hang under “gravity”

HW 3 due today I will not take any HW after 1:15pm Q1, Q2 and Q3 in different piles I will not take any HW after 1:15pm

Rest of Today’s agenda Prove n vertex tree has n-1 edges Algorithms for checking connectivity

Checking by inspection

What about large graphs? Are s and t connected?

Brute-force algorithm? List all possible vertex sequences between s and t nn such sequences Check if any is a path between s and t

Algorithm motivation all

Distance between u and v Length of the shortest length path between u and v Distance between RM and BO? 1

Questions?

Breadth First Search (BFS) Is s connected to t? Build layers of vertices connected to s Lj : all nodes at distance j from s L0 = {s} Assume L0,..,Lj have been constructed Lj+1 set of vertices not chosen yet but are connected to Lj Stop when new layer is empty

Exercise for you Prove that Lj has all nodes at distance j from s

BFS Tree BFS naturally defines a tree rooted at s Add non-tree edges Lj forms the jth “level” in the tree u in Lj+1 is child of v in Lj from which it was “discovered” 1 7 1 L0 2 3 L1 2 3 8 4 5 4 5 7 8 L2 6 6 L3